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We are tasked with solving the equation:

\[
\frac{\sqrt{5} - 1}{\sin x} + \frac{\sqrt{10} + 2\sqrt{5}}{\cos x} = 8
\]

We are looking for \( x \) in the interval \( \left(0, \frac{\pi}{2}\right) \).

### Step 1: Let’s denote:
- \( a = \frac{\sqrt{5} - 1}{\sin x} \)
- \( b = \frac{\sqrt{10} + 2\sqrt{5}}{\cos x} \)

The equation becomes:
\[
a + b = 8
\]

### Step 2: Use trigonometric identities
The values of sine and cosine are restricted by the interval \( \left(0, \frac{\pi}{2}\right) \), so we will explore potential exact values for \( x \) based on the options provided in the question.

The four options are:
- \( \frac{\pi}{10} \)
- \( \frac{3\pi}{10} \)
- \( \frac{9\pi}{10} \)
- \( \frac{7\pi}{10} \)

Let's calculate the values for each option and substitute back into the equation to check which value of \( x \) satisfies the equation.

I'll calculate this step-by-step.After evaluating the expression for each option:

- For \( x = \frac{\pi}{10} \), the result is approximately 12.03.
- For \( x = \frac{3\pi}{10} \), the result is approximately 14.52.
- For \( x = \frac{7\pi}{10} \), the result is approximately -11.46.
- For \( x = \frac{9\pi}{10} \), the result is approximately -4.03.

None of these results satisfy the equation \( \frac{\sqrt{5} - 1}{\sin x} + \frac{\sqrt{10} + 2\sqrt{5}}{\cos x} = 8 \). Since the interval is \( \left( 0, \frac{\pi}{2} \right) \), the most appropriate approach would be to explore a recalculation or the use of trigonometric approximations.

Would you like to proceed with checking other values or refine the analysis further? Here are some relative questions you may consider:

1. How can we refine our approach to check the exact solution using trigonometric identities?
2. Are there any specific trigonometric approximations you would like to explore further?
3. Should we plot the function to see where it crosses the value 8?
4. Would numerical methods for solving \( x \) be helpful in this case?
5. How would using inverse trigonometric functions assist in finding a more accurate solution?

**Tip:** Sometimes, complex trigonometric equations can be solved more efficiently with numerical methods or graphing techniques.

The value of x in (0, π/2) satisfying the equation: sqrt(5) - 1 / sin(x) + sqrt(10) + 2sqrt(5) / cos(x) = 8 is?

Learn how to solve the trigonometric equation involving sin(x) and cos(x) in the interval (0, π/2). This problem involves advanced trigonometric concepts and equation-solving methods, ideal for high school students in Grades 11-12.

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